Estimating Limit Values from Graphs - AP Calculus BC
Card 1 of 30
What does the graph of $f(x)$ show if $\text{lim}_{x \to c} f(x) \neq f(c)$?
What does the graph of $f(x)$ show if $\text{lim}_{x \to c} f(x) \neq f(c)$?
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A point discontinuity at $x = c$. When limit exists but doesn't equal function value at that point.
A point discontinuity at $x = c$. When limit exists but doesn't equal function value at that point.
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Identify the limit of $f(x)$ as $x$ approaches 2 from the right on a graph.
Identify the limit of $f(x)$ as $x$ approaches 2 from the right on a graph.
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The $y$-value that $f(x)$ approaches as $x$ approaches 2 from the right. Read the $y$-coordinate where the curve heads as $x$ nears 2 from positive side.
The $y$-value that $f(x)$ approaches as $x$ approaches 2 from the right. Read the $y$-coordinate where the curve heads as $x$ nears 2 from positive side.
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Estimate $\text{lim}_{x \to 0} f(x)$ from a graph with a vertical asymptote at $x = 0$.
Estimate $\text{lim}_{x \to 0} f(x)$ from a graph with a vertical asymptote at $x = 0$.
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The limit does not exist. Vertical asymptotes cause limits to not exist.
The limit does not exist. Vertical asymptotes cause limits to not exist.
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What is the implication if $\text{lim}{x \to c} f(x)$ exists but $\text{lim}{x \to c} f(x) \neq f(c)$?
What is the implication if $\text{lim}{x \to c} f(x)$ exists but $\text{lim}{x \to c} f(x) \neq f(c)$?
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A removable discontinuity at $x = c$. Limit exists but function is not continuous at that point.
A removable discontinuity at $x = c$. Limit exists but function is not continuous at that point.
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If a graph has a vertical asymptote at $x = 1$, what is $\text{lim}_{x \to 1} f(x)$?
If a graph has a vertical asymptote at $x = 1$, what is $\text{lim}_{x \to 1} f(x)$?
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The limit does not exist. Vertical asymptotes mean function goes to $\pm\infty$, so limit doesn't exist.
The limit does not exist. Vertical asymptotes mean function goes to $\pm\infty$, so limit doesn't exist.
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Which notation represents the limit of $f(x)$ as $x$ approaches 0 from the left?
Which notation represents the limit of $f(x)$ as $x$ approaches 0 from the left?
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$\text{lim}_{x \to 0^-} f(x)$. The minus sign indicates approaching from the negative (left) side.
$\text{lim}_{x \to 0^-} f(x)$. The minus sign indicates approaching from the negative (left) side.
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Estimate $\text{lim}_{x \to -\text{infinity}} f(x)$ if the graph approaches 0.
Estimate $\text{lim}_{x \to -\text{infinity}} f(x)$ if the graph approaches 0.
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- Read where graph heads as $x$ decreases without bound.
- Read where graph heads as $x$ decreases without bound.
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What is $\text{lim}_{x \to \text{infinity}} f(x)$ if the graph levels off at 7?
What is $\text{lim}_{x \to \text{infinity}} f(x)$ if the graph levels off at 7?
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- Horizontal asymptotes show end behavior as $x \to \infty$.
- Horizontal asymptotes show end behavior as $x \to \infty$.
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What is $\text{lim}_{x \to 3} f(x)$ if the graph shows a hole at $x = 3$?
What is $\text{lim}_{x \to 3} f(x)$ if the graph shows a hole at $x = 3$?
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The $y$-value the graph approaches as $x$ approaches 3. Holes don't affect limits; read where the curve would go.
The $y$-value the graph approaches as $x$ approaches 3. Holes don't affect limits; read where the curve would go.
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What does it mean if $\text{lim}_{x \to c} f(x) = L$?
What does it mean if $\text{lim}_{x \to c} f(x) = L$?
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As $x$ approaches $c$, $f(x)$ approaches $L$. Standard limit notation expressing function behavior near point $c$.
As $x$ approaches $c$, $f(x)$ approaches $L$. Standard limit notation expressing function behavior near point $c$.
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Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that approaches $y = 0$.
Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that approaches $y = 0$.
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- Horizontal asymptote shows end behavior approaching zero.
- Horizontal asymptote shows end behavior approaching zero.
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What does $\text{lim}_{x \to 0^-} f(x) = 3$ indicate about $f(x)$?
What does $\text{lim}_{x \to 0^-} f(x) = 3$ indicate about $f(x)$?
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$f(x)$ approaches 3 as $x$ approaches 0 from the left. Left-hand limit describes approach from negative direction.
$f(x)$ approaches 3 as $x$ approaches 0 from the left. Left-hand limit describes approach from negative direction.
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What is implied if $\text{lim}_{x \to \text{infinity}} f(x) = L$?
What is implied if $\text{lim}_{x \to \text{infinity}} f(x) = L$?
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The graph of $f(x)$ approaches $y = L$ as $x$ increases. Describes horizontal asymptote behavior at infinity.
The graph of $f(x)$ approaches $y = L$ as $x$ increases. Describes horizontal asymptote behavior at infinity.
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What is indicated by $\text{lim}_{x \to c} f(x) = L$ on a continuous graph?
What is indicated by $\text{lim}_{x \to c} f(x) = L$ on a continuous graph?
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The function value $f(c) = L$. Continuous functions have limits equal to function values.
The function value $f(c) = L$. Continuous functions have limits equal to function values.
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Estimate $\text{lim}_{x \to -3^+} f(x)$ if the graph shows $f(x)$ approaching 5.
Estimate $\text{lim}_{x \to -3^+} f(x)$ if the graph shows $f(x)$ approaching 5.
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- Right-hand limit reads $y$-value from positive approach.
- Right-hand limit reads $y$-value from positive approach.
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Determine $\text{lim}_{x \to 0} f(x)$ if graph shows $f(x)$ approaches 3 from both sides.
Determine $\text{lim}_{x \to 0} f(x)$ if graph shows $f(x)$ approaches 3 from both sides.
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- When both one-sided limits equal the same value.
- When both one-sided limits equal the same value.
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Estimate $\text{lim}_{x \to 4^-} f(x)$ from a graph showing $f(x)$ approaches 2.
Estimate $\text{lim}_{x \to 4^-} f(x)$ from a graph showing $f(x)$ approaches 2.
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- Left-hand limit reads $y$-value approached from negative side.
- Left-hand limit reads $y$-value approached from negative side.
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Determine the limit if $\text{lim}{x \to 0^-} f(x) = -4$ and $\text{lim}{x \to 0^+} f(x) = 4$.
Determine the limit if $\text{lim}{x \to 0^-} f(x) = -4$ and $\text{lim}{x \to 0^+} f(x) = 4$.
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The limit does not exist. Unequal one-sided limits mean no two-sided limit exists.
The limit does not exist. Unequal one-sided limits mean no two-sided limit exists.
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Estimate $\text{lim}_{x \to 7} f(x)$ if the graph shows a removable discontinuity at $x = 7$.
Estimate $\text{lim}_{x \to 7} f(x)$ if the graph shows a removable discontinuity at $x = 7$.
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The $y$-value the graph approaches, not the point at the discontinuity. Removable discontinuities don't affect limit values.
The $y$-value the graph approaches, not the point at the discontinuity. Removable discontinuities don't affect limit values.
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Find $\text{lim}_{x \to 2} f(x)$ if the graph is continuous at $x = 2$ and $f(2) = 8$.
Find $\text{lim}_{x \to 2} f(x)$ if the graph is continuous at $x = 2$ and $f(2) = 8$.
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- For continuous functions, limit equals function value.
- For continuous functions, limit equals function value.
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Identify $\text{lim}_{x \to 1^+} f(x)$ from a graph approaching value 4.
Identify $\text{lim}_{x \to 1^+} f(x)$ from a graph approaching value 4.
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- Right-hand limit reads $y$-value approached from positive side.
- Right-hand limit reads $y$-value approached from positive side.
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Estimate $\text{lim}_{x \to 2} f(x)$ from a graph where $f(x)$ approaches $-3$.
Estimate $\text{lim}_{x \to 2} f(x)$ from a graph where $f(x)$ approaches $-3$.
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-3. Read the $y$-value the graph approaches at $x = 2$.
-3. Read the $y$-value the graph approaches at $x = 2$.
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What is the condition for a limit to exist at $x = c$?
What is the condition for a limit to exist at $x = c$?
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$\text{lim}{x \to c^-} f(x) = \text{lim}{x \to c^+} f(x)$. Both one-sided limits must exist and be equal.
$\text{lim}{x \to c^-} f(x) = \text{lim}{x \to c^+} f(x)$. Both one-sided limits must exist and be equal.
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Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that levels off at $y = 9$.
Estimate $\text{lim}_{x \to \text{infinity}} f(x)$ from a graph that levels off at $y = 9$.
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- Horizontal asymptote at $y = 9$ as $x$ increases.
- Horizontal asymptote at $y = 9$ as $x$ increases.
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Estimate $\text{lim}_{x \to 3^-} f(x)$ if the graph shows $f(x)$ approaching 6.
Estimate $\text{lim}_{x \to 3^-} f(x)$ if the graph shows $f(x)$ approaching 6.
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- Left-hand limit reads $y$-value from negative approach.
- Left-hand limit reads $y$-value from negative approach.
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Estimate $\text{lim}_{x \to 5} f(x)$ if graph approaches different values from left and right.
Estimate $\text{lim}_{x \to 5} f(x)$ if graph approaches different values from left and right.
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The limit does not exist. Different one-sided limits mean the two-sided limit doesn't exist.
The limit does not exist. Different one-sided limits mean the two-sided limit doesn't exist.
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If $\text{lim}{x \to c^-} f(x) \neq \text{lim}{x \to c^+} f(x)$, what can be said about $\text{lim}_{x \to c} f(x)$?
If $\text{lim}{x \to c^-} f(x) \neq \text{lim}{x \to c^+} f(x)$, what can be said about $\text{lim}_{x \to c} f(x)$?
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The limit does not exist. Two-sided limit requires both one-sided limits to be equal.
The limit does not exist. Two-sided limit requires both one-sided limits to be equal.
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Estimate $\text{lim}_{x \to -2} f(x)$ from the graph if $f(x)$ approaches 5.
Estimate $\text{lim}_{x \to -2} f(x)$ from the graph if $f(x)$ approaches 5.
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- Read the $y$-value the graph approaches at $x = -2$.
- Read the $y$-value the graph approaches at $x = -2$.
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What is implied if $\text{lim}_{x \to \text{infinity}} f(x) = L$?
What is implied if $\text{lim}_{x \to \text{infinity}} f(x) = L$?
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The graph of $f(x)$ approaches $y = L$ as $x$ increases. Describes horizontal asymptote behavior at infinity.
The graph of $f(x)$ approaches $y = L$ as $x$ increases. Describes horizontal asymptote behavior at infinity.
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Determine $ \lim_{x \to 0} f(x) $ if graph shows $f(x)$ approaches 3 from both sides.
Determine $ \lim_{x \to 0} f(x) $ if graph shows $f(x)$ approaches 3 from both sides.
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- When both one-sided limits equal the same value.
- When both one-sided limits equal the same value.
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